Study of digital topology concepts like hyperspaces and function graphs.
problem Adapting classical topology concepts to digital topology.
method Define digital hyperspaces and function graphs, study their properties.
result Some relationships and graphical properties of digital hyperspaces and function graphs.
Critiques incorrect fixed point assertions in digital topology.
problem Incorrect or incorrectly proven fixed point assertions in digital topology.
method Critical review of existing assertions.
result Identifies and critiques incorrect fixed point assertions.
Corrects incorrect assertions about fixed points in digital topology.
problem Incorrect or incorrectly proven assertions about fixed points in digital metric spaces.
method Analysis of existing assertions and proofs.
result Identifies and corrects errors in published assertions.
Find limiting sets for digital cones and suspensions.
problem Digital topology cone and suspension constructions.
method Identify (m, n)-limiting sets, especially (0, 0)-freezing sets.
result Discover (0, 0)-limiting sets for digital cones and suspensions.
The paper highlights issues in fixed point claims in digital topology.
problem Flaws in published assertions about fixed points in digital metric spaces.
method Continues a series of studies examining these flaws.
result Identifies and discusses problems in fixed point claims.
Study minimal freezing sets in convex digital disks.
problem Finding minimal freezing sets in convex digital disks.
method Showed how to find minimal freezing sets for convex disks in digital plane.
result Found minimal freezing sets for convex disks in digital plane.
Incorrect fixed point assertions in digital topology are discussed.
problem Incorrect or poorly stated fixed point assertions in digital topology.
method Discussion of problematic publications in digital metric spaces.
result Clarification of incorrect fixed point assertions.
Incorrect fixed point assertions in digital topology are discussed.
problem Incorrect, incorrectly proven, or trivial fixed point assertions in digital topology.
method Continues earlier work on identifying and critiquing bad fixed point assertions.
result Clarifies the nature and extent of incorrect fixed point assertions in digital topology.
The paper highlights issues with fixed point claims in digital images.
problem Flaws in published assertions about fixed points in digital images.
method Continues a series of studies examining digital topology.
result Identifies and discusses problems with fixed point claims.
Incorrect fixed point results in digital topology are debunked.
problem Incorrect fixed point results in digital topology.
method Mimicking classical topology tools.
result Many recent fixed point results are incorrect or trivial.
Incorrect fixed point results in digital topology are corrected.
problem Incorrect fixed point results in digital topology.
method Analysis of recent papers in digital topology.
result Corrected incorrect conclusions in fixed point results.
The paper addresses flaws in fixed point assertions for digital images.
problem Deficiencies in previously published works on fixed point assertions for digital images.
method Continues a series of studies to identify and rectify issues in fixed point assertions.
result Identifies and corrects flaws in fixed point assertions for digital images.
Fixed point assertions in digital topology are often incorrect or poorly stated.
problem Fixed points in digital metric spaces
method Discussing publications with bad assertions
result Identifying and correcting errors in fixed point assertions
The study examines properties of digital images using various adjacencies.
problem Properties of Cartesian products of digital images.
method Various adjacencies used to study digital images.
result Properties of digital images studied using adjacencies.
The paper corrects and improves previous assertions in digital topology.
problem Incorrect or insufficiently proven assertions in digital topology.
method Review and correction of existing assertions.
result Improved and corrected assertions in digital topology.
Study restrictions on digitally continuous functions and their effects.
problem Understanding effects of restrictions on digitally continuous functions.
method Analyzing digitally continuous functions and their modifications.
result Analogous result for topological spaces derived from digitally continuous functions.
The paper corrects and improves previous assertions in digital topology.
problem Incorrect or poorly proven assertions in digital topology.
method Review and correction of existing assertions.
result Improved and corrected assertions in digital topology.
Digital trees have approximate fixed point property, and conditions for products are explored.
problem Conditions for the approximate fixed point property in digital tree products.
method Analyzes digital trees and their products, explores conditions for the AFPP.
result Conditions are found for the AFPP in digital tree products.
The paper constructs graph models for n-dimensional manifolds.
problem Creating digital models of n-dimensional manifolds.
method Constructing graph models using LCL collections of n-cells.
result Digital models retain topological properties of continuous manifolds.
We study properties of shy maps in digital topology.
Study fixed points in digital images, introducing new invariants.
problem Understanding properties of digital images through fixed points.
method Introduce new invariants and freezing/cold sets to analyze fixed point sets.
result Existence of fixed point sets restricts maps on their complements.
Pruned neural networks learn digital circuits with 99% weight reduction.
problem Efficiently train deep neural networks with minimal weights.
method Constrained binarized networks to zero or one weights.
result Pruned networks achieve similar performance to standard networks with 99% weight reduction.
New tools for constructing fixed point sets in digital topology.
problem Constructing fixed point sets in digital topology.
method Defining excludable points and articulation points, and showing their exclusion from freezing sets.
result Excludable points and articulation points can be excluded from all freezing sets.
Topology speeds up and improves CNNs for digit recognition.
problem Improving and speeding up deep learning models for digit recognition.
method Topological data analysis applied to CNNs to modify computations.
result Topology improves CNNs' performance and generalization.
We give an answer to the question given by T.Y.Kong in his article "Can 3-D Digital Topology be Based on Axiomatically Defined Digital Spaces?" In this article he asks the question, if so called "good pairs" of neighborhood relations can be found on the set Z^n such that the existence of digital manifolds of dimension …
TDA classifies MNIST digits with reduced feature set.
problem Classifying MNIST digits using machine learning.
method Persistent homology for feature generation and classification.
result 5x reduction in feature set size with similar accuracy.
Corrects errors in Hans' pseudocovering spaces paper.
problem Mathematical errors and citation issues in Hans' pseudocovering spaces paper.
method Identifies and corrects errors in Hans' previous work.
result Addresses mathematical and citation errors in Hans' pseudocovering spaces paper.
3D self-affine tiles with specific digit sets have boundary homeomorphic to a 2-sphere.
problem Characterizing 3D self-affine tiles with collinear digit sets whose boundary is a sphere.
method Using lattice tiling combinatorics and topological properties of spheres, the paper characterizes such tiles.
result The boundary of these tiles is homeomorphic to a 2-sphere under certain conditions.
The paper presents a new set of axioms of digital topology, which are easily understandable for application developers. They define a class of locally finite (LF) topological spaces. An important property of LF spaces satisfying the axioms is that the neighborhood relation is antisymmetric and transitive. Therefore any…
The study examines two types of fractals and their topological properties.
problem Analyzing the topological properties of self-similar fractals with a shifting parameter.
method Detailed discussion and proof for disk-likeness and connectivity.
result Conditions for Tε to be quasi-periodic and connected. This study redefines probability for finite outcomes using axioms and examples.
problem Defining probability for finite outcomes and preserving information.
method Developed three axioms for relative probability functions and provided examples and a system for their composition.
result Proved the topological closure of the relative probability space, preserving information under limits.
We construct an elementary, combinatorial kind of topological quantum field theory, based on curves, surfaces, and orientations. The construction derives from contact invariants in sutured Floer homology and is essentially an elaboration of a TQFT defined by Honda--Kazez--Matic. This topological field theory stores inf…
Shy maps preserve path-connectedness in continuous functions.
problem Understanding properties of continuous functions in topological spaces.
method Defining shy maps as continuous functions with specific path-connected inverse images.
result Every shy map onto a semilocally simply connected space induces a surjection of fundamental groups.
Given an integer n≥2 and a digit set D⊊0,1,...,n−12, there is a self-similar set F⊂R2 satisfying the set equation: F=(F+D)/n. We call such F a fractal square. By studying a periodic extension H=F+Z2, we classify F into three types accordi…
Self-affine tiles homeomorphic to a ball proven for a specific digit set.
problem Topology of self-affine tiles with collinear digit sets.
method Proving homeomorphism to a ball using integral self-affine tiles with collinear digit sets.
result A large class of integral self-affine tiles with collinear digit sets is homeomorphic to a closed 3-dimensional ball.
The paper uses topological concepts to analyze neural networks, revealing complex structure and dynamics.
problem Understanding the structure and dynamics of deep learning models.
method Topological dynamical systems, index theory, and computational homology.
result Neurons correspond to simplexes in a simplicial complex, and topological invariants can be computed.
ISOMORPH creates a digital twin for supply chain logistics, advancing time-series forecasting benchmarks.
problem Lack of public benchmarks for supply chain logistics time-series forecasting.
method Developed a digital twin simulator with interpretable parameters and modular topology, generating datasets and verifying conservation laws.
result Foundation models achieve MASE values exceeding public benchmarks at low-to-moderate horizons, supporting UQ.
Digital money could reduce germ spread during coronavirus.
problem Spreading of germs via paper money during coronavirus.
method Policy recommendations for mobile wallets, digital currencies, and data protection.
result Adopting digital money can help reduce germ spread.
Method recognizes chaotic regime in crypto markets.
problem Detecting critical transitions in crypto markets.
method Topological data analysis + k-means clustering.
result Early warning signals for crypto market crashes.
Corrects a false claim about a digital sphere model's contractibility.
problem Incorrect claim about MSS_18's 18-contractibility.
method Analyzes digital image MSS_18 as a digital model of S^2.
result Shows MSS_18 is 18-contractible.
This paper optimizes cybersecurity resource allocation in networks with heterogeneous attacker and defender valuations.
problem Optimizing cybersecurity resource allocation in networks with heterogeneous attacker and defender valuations.
method Combining strategic behavior of players with contagion dynamics, a method is extended to determine optimal resource allocation based on simple network metrics weighted by risk profiles.
result The asymmetry between attacker and defender valuations drives optimal attack and defense strategies, shaping system resilience.
Study AFPP of unions of convex digital disks in 2D.
problem Conditions for AFPP of union of convex disks in digital plane.
method Use results from [6] to analyze AFPP.
result Conditions for AFPP of union of convex disks.
Study on cold and freezing sets in digital images.
problem Properties of cold sets in digital images.
method Analysis of properties and relationships between cold and freezing sets.
result Examined relationships between cold and freezing sets.
Study convexity and AFPP in digital images.
problem Relationship between convexity and AFPP in digital images.
method Examined in Z^2 digital images.
result Relationship between convexity and AFPP in digital images.
Han discusses variants of digital covering maps and their equivalences.
problem Han's paper lacks thorough discussion on variants and their equivalences.
method Examined several variants of digital covering maps and compared their equivalences.
result Found several equivalences among the variants of digital covering maps.
Examines how irreducibility and rigidity affect digital images.
problem Understanding interactions between irreducibility and rigidity in digital images.
method Analyzes Cartesian products, wedges, and cold and freezing sets.
result Interactions between irreducibility and rigidity in digital images.
A new method streamlines digital payment programming using smart contracts.
problem High costs and security challenges in programming smart contracts for digital payments.
method Transforming digital currencies into token streams and using configurable templates to generate specialized smart contracts.
result Reduces payment programming costs and enhances security, self-enforcement, adaptability, and controllability.
New framework explains leading digit patterns without probabilistic assumptions.
problem Explaining leading digit distributions without relying on probabilistic models.
method Shift-invariant functional equation and affine-plus-periodic formulas.
result Unified mathematical foundation for understanding digit distributions.